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Tutorial

Properties of Radicals

Cautions when Applying the Properties

Applying the Properties of Radicals

Properties of Radicals

There are several properties of radicals we can apply to simplify expressions involving radicals. The following properties are generally true whenever n is greater than 1, and a and b are both positive real numbers:

and

Product Property:

Quotient Property:

Fractional Exponents:

Cautions when Apply the Properties

Avoid these common errors when applying properties of radicals:

The properties of radicals only apply to factors; they do not apply to terms. For example, we can use the product property of radicals to break into two radicals because 5 • 3 = 15. However, we cannot break into

We can only bring an exponent outside of a radical if it applies to everything underneath the radical. For example, we can rewrite as because the exponent of 2 applied to everything underneath the radical. However, This is because the exponent of 2 applies only to the x, not the 16. (We could rewrite the expression as because 16 = 42)

Taking the odd-root of a negative number leads to a real number solution, because a negative value raised to an odd exponent is negative. However, taking the even-root of a negative value leads to a non-real solution, because a negative value raised to an even exponent is never negative.

Applying the Properties of Exponents

When we recognize products, quotients, and powers with radicals, we can apply the properties of radicals to simplify the expression. This is shown in the examples below: